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Groupoids and C*-algebras for categories of paths

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arxiv 1111.6924 v4 pith:OL33PKDY submitted 2011-11-29 math.OA

classification math.OA
keywords algebrasgraphspathsaperiodicitycancellationcategoriescombinatorialconcatenation
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In this paper we describe a new method of defining C*-algebras from oriented combinatorial data, thereby generalizing the constructions of algebras from directed graphs, higher-rank graphs, and ordered groups. We show that only the most elementary notions of concatenation and cancellation of paths are required to define versions of Cuntz-Krieger and Toeplitz-Cuntz-Krieger algebras, and the presentation by generators and relations follows naturally. We give sufficient conditions for the existence of an AF core, hence of the nuclearity of the C*-algebras, and for aperiodicity, which is used to prove the standard uniqueness theorems.

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  1. Subshift semigroups

    math.OA 2019-08 accept novelty 7.0 of 10

    For every subshift, the Matsumoto and Carlsen-Matsumoto C*-algebras are realized as groupoid C*-algebras from the inverse hull of the language semigroup, and this universal groupoid is amenable.

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